Let \(\eta _t\)  be the density of \(S_t\) . It follows from [1]} that
\(p_\alpha (t,x)=\int _0^\infty (2\pi s)^{-\frac{d}{2}}\text{\rm {e}}^{-\frac{|x|^2}{2s}}\eta _t(s)\text{\rm {d}}s\asymp t(t^{1/\alpha }+|x|)^{-d-\alpha }, ~(t,x)\in (0,\infty )\times \mathbb {R}^d.\) 
